Uniqueness in Calderón's problem for conductivities with unbounded gradient

Abstract

We prove uniqueness in the inverse conductivity problem for uniformly elliptic conductivities in $W^{s,p}(\Omega )$, where $\Omega \subset {\mathbb{R}}^n$ is Lipschitz, $3\le n\le 6$, and $s$ and $p$ are such that $W^{s,p}(\Omega )\not\subset W^{1,\infty }(\Omega )$. In particular, we obtain uniqueness for conductivities in $W^{1,n}(\Omega )$ ($n=3,4$). This improves on the result of the author and Tataru, who assumed that the conductivity is Lipschitz.

minor changes, to appear in CMP

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